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Because senators want to pass this bill with only a simple majority, it faces special restrictions

The House bill will have to go through what's known as the "Byrd Bath" in the Senate

Washington (CNN)The Obamacare repeal bill that passed the House Thursday moves next to the Senate where it faces daunting challenges because of the same ideological splits between conservative and moderate Republicans that nearly killed it in the House.

GOP leaders have set up a working group of senators across the ideological spectrum to try search for compromises that could unite enough Republicans to get the 51 votes needed for it to clear the chamber.

"There is a working group over here of Republicans with, you know, with a range of ideology that are working to see where we go with the bill when it comes across, said Sen. Bob Corker, R-Tennessee, in an interview on MSNBC. "And I think you're going to see very responsible, deliberate action on it. People are going to want to improve it. I don't see anyway he comes back in the form it comes. It's not because I have any specific criticism of it."

One of the biggest challenges the House bill faces is that in order for it to be eligible for simple majority vote -- instead of a 60-vote threshold usually required for major bills -- it must meet special requirements under budget reconciliation rules. One of the key demands is that it lower the deficit. It is not known if the latest iteration of the House bill will do that, especially after money was added to it to resolve differences between the hard right House Freedom Caucus and the moderate Tuesday Group.

The House bill will have to go through what's known as the "Byrd Bath," named after the late long-serving Democratic senator from West Virginia, Robert Byrd. It is the Senate Parliamentarian who makes those determinations.

"If the House bill doesn't pass the Byrd Rule, than we can modify it in the Senate and try to work out the differences in conference," said Sen. John Cornyn, R-Texas, referring to a conference committee of House and Senate members that could be formed to resolve differences between final House and Senate bills. "That's really the way legislation is supposed to work. I hope they pass it ... and then our job begins again."

Asked how long it will take the Senate to act on a bill, Cornyn acknowledged it won't be quick.

The person who must lead the talks and finally shape a compromise is Majority Leader Mitch McConnell, R-Kentucky. This week he hinted to reporters how difficult the task could be.

"When they send it over here it will be a real big challenge on the Senate side as well," he said.

Because the margins are so narrow, McConnell can't afford any defections so every GOP senator could have enormous sway. At the same time, for Republicans to meet their long-sought goal of repealing Obamacare, they must find ways to not be rigid in their demands.

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But even if they pass something, there is no assurance the House would accept a dramatically altered bill so the repeal effort could fail then.

"If it goes in a free market direction that lowers prices and it doesn't resemble the bill, that's fine," Rep. Dave Brat, R-Virginia, a member of the House Freedom Caucus told CNN. "But if it goes toward more central government, then no. The odds are low."

And Freedom Caucus Chairman Rep. Mark Meadows said that he's already talked to 14 senators about how to move the House bill on that side.